The 19-year-old has impressed so far in Germany with two assists and a goal in his eight league appearances.
"I want to win the Bundesliga and I want to win it with RB Leipzig. Why not?" he told Sky.
"We have the players, the quality. We have a great team, so there is no reason why we can't."
Burke, 19, could make his fifth appearance for Scotland in Friday's World Cup qualifier against old rivals England at Wembley.
He became the record signing for a Scottish player when RB Leipzig paid Nottingham Forest 15 million euros ($16.8m) from him in August.
Burke has impressed so far in Germany with two assists and a goal in his eight league appearances.
Leipzig team are level on 24 points with Bayern, second in the table only on goal difference, and are on a record-equalling run for a team in their first Bundesliga season after ten games unbeaten.
Director of sport Ralf Rangnick and coach Ralph Hasenhuettl have repeatedly shaken off the 'Bayern Hunters' tag the German media are eager to bestow upon RB Leipzig.
With Bayern on course for a fifth straight Bundesliga crown, Burke is the first RB player to openly talk of Leipzig winning the title and break the Bavarian giants' record run.
Leipzig, who are bankrolled by Austrian energy drinks giants Red Bull, were founded in 2009.
They earned their place in Germany's top flight with four promotions in seven years.
Leipzig have beaten sides likes Borussia Dortmund, Hamburg, Wolfsburg and Werder Bremen, but face big tests in the coming weeks.
After the international break, they are away to Bayer Leverkusen a week on Friday and play Schalke and Hertha Berlin before facing Bayern at Munich's Allianz Arena on December 20.
"It's an amazing feeling to be second in the Bundesliga at the moment," said Burke.
"The team is so close and I've bonded very well with the players.
"Many of them speak English so it is quite easy to adapt. They are a great bunch of boys and we're all striving for the same thing and want the same goal -- to do our best and to go into every game to win.
"Everyone here is like a family, everyone looks after everyone else."